Effects of confinement on the cubic bicontinuous phase with double-diamond symmetry (D) are studied. We focus on a slit geometry and on highly selective external walls. We show that the period of the solvation force in different systems containing amphiphilic molecules can be found on simple symmetry grounds. The simple symmetry-based predictions are verified by explicit calculations in a lattice model for oil-water-surfactant mixtures. The explicit results obtained within the model allow for a detailed description of the effects of the confinement on the structure of the D phase. The structure of the D phase with the face of the unit cell parallel to the external wall is shown and discussed for slits ranging from very narrow to very large ones.